schur¶
Purpose¶
Computes the real or complex Schur form of a square matrix with the option to sort the eigenvalues.
Format¶
-
{ S, Z } =
schur(A[, flag[, sort_type]])¶ - Parameters:
A (KxK matrix) – data
flag (string) –
Optional input, to control whether output should be in real or complex Schur form. Valid options include:
"complex""real"
sort_type (string) –
Optional input, specifying how to sort the eigenvalues. Options include:
1
”udi”
Absolute value of the eigenvalue less than 1.0. (Unit disk inside)
2
”udo”
Absolute value of the eigenvalue greater than or equal to 1.0. (Unit disk outside)
3
”lhp”
Value of the real portion of the eigenvalue less than 0. (Left hand plane)
4
”rhp”
Value of the real portion of the eigenvalue greater than 0. (Right hand plane)
5
”ref”
Real eigenvalues first. (Complex portion less than imagtol see remarks section)
6
”cef”
Complex eigenvalues first. (Complex portion greater than imagtol see remarks section)
- Returns:
S (KxK matrix) – Schur form.
Z (KxK matrix) – transformation matrix.
Examples¶
Real matrix with all real eigenvalues¶
// Create a 2 x 2 matrix
A = { 7 -2,
12 -5 };
// Calculate eigenvalues of 'A'
lambda = eig(A);
After the code above, lambda should equal:
4.4641
-2.4641
// Continuing with 'A' from above
{ S, Z } = schur(A);
Now S and Z should equal:
S = 4.4641 -14.000 Z = 0.6193 -0.7852
0 -2.4641 0.7852 0.6193
Real matrix with some complex eigenvalues¶
// Create a 3 x 3 matrix
A = { 1 -4 -1,
3 -1 9,
-9 1 -2 };
// Calculate real schur form, with complex eigenvalues
// stored as 2 x 2 blocks on the diagonal
{ S_r, Z_r } = schur(A, "real");
// Calculate complex schur form
{ S_c, Z_c } = schur(A, "complex");
After the code above:
S_r = -7.2208 8.6875 -1.7726
-4.4007 -1.1773 1.5989
0 0 6.3981
Z_r = -0.1304 0.7105 -0.6915
0.6913 0.5651 0.4502
-0.7107 0.4193 0.5649
S_c = -4.1991+5.3945i -2.5084+6.9720i 1.0168-0.7763i
0+0i -4.1991-5.3945i 1.9825-0.3630i
0+0i 0+0i 6.3981+0i
Z_c = 0.3275 - 0.1759i 0.5326 + 0.3160i -0.6915
0.1102 - 0.7700i 0.0023 + 0.4385i 0.4502
0.3132 + 0.3984i 0.6502 + 0.0373i 0.5649
Complex matrix with sorted eigenvalues¶
// Create a 3 x 3 complex matrix
A = { -1.9615 + 0.4382i 0.0655 + 0.6913i -1.1424 + 0.1997i,
0.1244 + 0.3783i -0.2821 + 0.0588i 0.4854 + 0.4700i,
1.1271 + 0.7045i -1.5245 - 0.9966i 1.4969 + 0.4450i };
// Place eigenvalues in unit circle at top-left
{ S, Z } = schur(A, "complex", "udi");
After the code above:
-0.3548 + 0.8005i 2.4873 - 0.4942i -1.3144 + 0.7286i
S = 0.0000 1.0504 - 0.5581i 0.1763 + 0.7846i
0.0000 0.0000 -1.4423 + 0.6996i
0.3692 - 0.2393i -0.0144 + 0.2838i 0.5890 - 0.6155i
Z = -0.3907 + 0.2625i 0.5407 - 0.5251i 0.4161 - 0.1930i
-0.7530 - 0.1336i -0.5813 + 0.1154i 0.2225 - 0.1201i
Remarks¶
If a real matrix is passed in without a flag variable, the real Schur form will be returned. If a complex matrix is passed in without a flag variable, GAUSS will check to see if any of the imaginary elements are greater than imagtol (2.23e-16 by default). If any imaginary elements are greater than imagtol, the complex Schur form will be calculated, otherwise the real Schur form will be returned. If a real flag is passed in with a complex matrix, the flag will be ignored and the complex Schur factorization will be returned.
The real Schur form is an upper quasi-triangular matrix, that is, it is
block triangular where the blocks are 2x2 submatrices which correspond
to complex eigenvalues of A. If A has no complex eigenvalues, S will be
strictly upper triangular. To convert the real Schur form of S to the
complex Schur form, use the Run-Time Library function schtoc().
Z is an orthogonal matrix that transforms A into S and vice versa. Thus
S = Z'*A*Z;
and since Z is orthogonal,
A = Z*S*Z';